What does the term sampling distribution mean conceptually and rigorously? Sorry if this is basic, but I was reviewing hypothesis testing after 6 years and came across khan academy's video explaining it. At some point at minute 4.03 the speaker says "sampling distribution", what does he mean rigorously with that? From the spoken video and what he write it seems that he mean the distribution of the sample mean. i.e. if we have i.i.d. random variable $X_i$ and we define a new r.v.:
$$ M_m = \frac{1}{m} \sum^m_{i=1} X_i $$
then it seems that he refers to the sampling distribution as the distribution of $M_m$. It seems confusing because the sampling distribution for me in my head should be the distribution from where we obtain samples, i.e. the distribution of $X_i$ which is of  course the population distribution. 
I know this is probably mostly a conceptual question but why does he refer to the sampling distribution as $M_m$? It seems weird because we obtain samples from the population not from the distribution according to $M_m$. I think I am probably wrong so I wanted to understand why.  
 A: You are correct in your understanding. However, I can try to give some justification of why this term is used the way it is.
The "population" $X$ is an abstract concept, represented by a hypothetical distribution $p[X=x]$ which in principle assigns some probability to each numeric value $x$ that the random variable $X$ might take on. This population distribution is essentially never known. But we can try to learn about this distribution from measurements $X_1,\ldots,X_n$ which constitute a "sample" of the population.
Of course any statistic we compute from a sample, $Y=f[X_{1:n}]$, (e.g. $Y=\bar{X}$) is itself a random variable. The important point is that the "population" distribution of $Y$ is now a function of the population distribution $p[X=x]$ but also depends on the sampling scheme. For example $p[Y=y]$ will change if we have a different sample size $n$, or if we have a biased sampling scheme (e.g. due to detection tolerance).
If we just knew that $Y$ is some random variable, then we might refer to its distribution as a "population" distribution (or just "distribution", more casually). But if we know that $Y$ is actually some statistic computed from a sample of $X$, then we would refer to its distribution as a "sampling" distribution.
Hopefully this was not more confusing!
